Question

The following table contains the measurements of the key length dimension from a fuel injector. These samples of size five were taken at one-hour intervals.

Construct a three-sigma __ -chart and R-chart (use Exhibit 13.7) for the length of the fuel injector. What can you say about this process?

Short answer

The X-bar graph demonstrates how the mean, as well as average, varies over time, whereas the R chart depicts how the spectrum of subgroups evolves.

Step-by-step solution

Consider the data

Given data is as follows:

The sample mean and range are calculated as follows:

The Sample mean for set 1 = Sum of all the samples for set 1/Number of samples in set 1 =2.47/5 = 0.49

The Range for day 1 = Maximum value of all the samples in set 1 - Minimum value of all samples in set 1 = 0.511 -0.481 = 0.03

The details for all the sets are as follows:

Calculations for UCL, LCL, and X bar chart are as follows

The calculations for UCL, LCL, and X bar chart are as follows:

The ith subgroup mean is calculated using;

The grand mean (x-bar-bar) is calculated as follows:

Also, the following is another formula for the calculation of the grand mean(x-bar-bar):

The range of the above data is calculated using the formula below:

We also have the following A2, D3, and D4 values for a sample size of n = 5. For the chosen sample size (as determined by the table of constants):

The following are the calculations for UCL, LCL, as well as centerline for the X-bar chart

The R chart's UCL, LCL, and centerline calculations are as follows.

The x-bar and range computed values are within the UCL as well as LCL limits determined previously for the x-bar and r-chart, indicating that the process is statistically controlled.